Pulsars as gamma ray sources: Nebular shocks and magnetospheric gaps

I summarize the results of recent research on the structure and particle acceleration properties of relativistic shock waves in which the magnetic field is transverse to the flow direction in the upstream medium, and whose composition is primarily electrons and positrons with an admixture of heavy ions. Shocks which contain heavy ions that are a minority constituent by number but which carry most of the energy density in the upstream medium put 20% of the flow energy into a nonthermal population of pairs downstream, whose distribution in energy space is N(E) E ^-2, where N(E)dE is the number of particles with energy between E and E+dE. Synchrotron maser activity in the shock front, stimulated by the quasi-coherent gyration of the whole particle population as the plasma flowing into the shock reflects from the magnetic field in the shock front, provides the mechanism of thermalization and non-thermal particle acceleration. The maximum energy achievable by the pairs is _± m _± c ² = m _i c ² ₁/Z _i, where ₁ is the Lorentz factor of the upstream flow and Z _i is the atomic number of the ions. The shock's spatial structure contains a series of overshoots in the magnetic field, regions where the gyrating heavy ions compress the magnetic field to levels in excess of the eventual downstream value. These overshoots provide a new interpretation of the structure of the inner regions of the Crab Nebula, in particular of the wisps, surface brightness enhancements near the pulsar. The wisps appear brighter because the small Larmor radius pairs are compressed and radiate more efficiently in the regions of more intense magnetic field. This interpretation suggests that the structure of the shock terminating the pulsar's wind in the Crab Nebula is spatially resolved, and allows one to measure ₁ 4 × 10⁶, the upstream magnetic field B ₁ to be 3 × 10^-5 Gauss, as well as to show that the total ion flow is 3 × 10³⁴ elementary charges/sec, in good agreement with the total current flow predicted by the early Goldreich and Julian (1969) model. The total pair outflow is shown to be about 5 × 10³⁷ pairs per second, in good agreement with the particle flux required to explain the nebular X—ray source.The energetics of particle acceleration within the magnetospheres of rotation powered pulsars and the consequences for pulsed gamma ray emission are also briefly discussed. The gamma ray luminosity above 100 MeV is shown to scale in proportion to _R ^1/2 , as is in accord with some of the simplest ideas about polar cap models. Models based on acceleration in the outer magnetosphere are also briefly discussed.     

The importance of anisotropic interstellar turbulence and molecular-cloud magnetic mirrors for galactic cosmic-ray propagation

Recent studies suggest that when magnetohydrodynamic (MHD) turbulence is excited by stirring a plasma at large scales, the cascade of energy from large to small scales is anisotropic, in the sense that small-scale fluctuations satisfy the inequality k k , where k and k are, respectively, the components of a fluctuations wave vector and to the background magnetic field. Such anisotropic fluctuations are very inefficient at scattering cosmic rays. Results based on the quasilinear approximation for scattering of cosmic rays by anisotropic MHD turbulence are presented and explained. The important role played by molecular-cloud magnetic mirrors in confining and isotropizing cosmic rays when scattering is weak is also discussed.     

Recent results on the parameters of the interstellar helium from the ULYSSES/GAS experiment

Velocity and direction of the flow of the interstellar helium and its temperature and density have been determined from the measurements of the ULYSSES/GAS experiment for two different epochs: during the in-ecliptic path of ULYSSES, representing solar maximum conditions, and during the south to the north pole transition (11/94-6/95), close to the solar minimum conditions. Within the improved error bars the values are consistent with results published earlier.The determination of the density n of the interstellar helium at the heliospheric boundary from observations in the inner solar system requires knowledge about the loss processes experienced by the particles on their way to the observer. The simultaneous observation of the helium particles arriving on direct and indirect orbits at the observer provides a tool to directly determine the effects of the loss processes assumed to be predominantly photoionization and — for particles travelling close to the Sun — electron impact ionization by high-energy solar wind electrons.Such observations were obtained with the ULYSSES/GAS instrument in February 1995, before the spaceprobe passed its perihelion. From these measurements values for the loss rates and the interstellar density could be derived. Assuming photoionization to be the only loss process reasonable fits to the observations were obtained for an ionization rate = 1.1 · 10^–7 s^–1 and a density n 1.7 · 10^–2 cm^–3. Including, in addition, electron impact ionization, a photoionization = 0.6 · 10^–7 s^–1 was sufficient to fit both observations, resulting in a density n 1.4 · 10^–2 cm^–3.On leave from Space Research Centre, Warsaw, Poland.     

A New Mechanism of Coronal Heating

The interaction between network magnetic fields and emerging intranetwork fields may lead to magnetic reconnection and microflares, which generate fast shocks with an Alfvén Mach number M _A<2. Protons and less abundant ions in the solar corona are then heated and accelerated by fast shocks. Our study of shock heating shows that (a) the nearly nondeflection of ion motion across the shock ramp leads to a large perpendicular thermal velocity (v _th), which is an increasing function of the mass/charge ratio; (b) the heating by subcritical shocks with 1.1 M_A 1.5 leads to a large temperature anisotropy with T/T 50 for O⁵⁺ ions and a mild anisotropy with T_/T 1.2 for protons; (c) the large perpendicular thermal velocity of He⁺⁺ and O⁵⁺ ions can be converted to the radial outflow velocity (u) in the divergent coronal field lines; and (d) the heating and acceleration by shocks with 1.1 M_A 1.5 can lead to u(O⁵⁺) v _th(O⁵⁺) 460 km s^–1 for O⁵⁺ ions, u(He⁺⁺) v _th(He⁺⁺) 360 km s^–1 for He⁺⁺ ions, and u(H⁺) v _th(H⁺) 240 km s^–1 for protons at r=3–4 R . Our results can explain recent SOHO observations of the heating and acceleration of protons and heavier ions in the solar corona.     

Some results of ionospheric investigations by means of coherent radiowaves emitted by satellites

In this paper we discuss theoretical expressions, determining the difference of Doppler shifts of various coherent radiowave frequencies emitted by a radiator moving in the ionosphere or interplanetary medium. The rotating Doppler effect (Faraday effect) caused by the Doppler shifts ±H of the ordinary and extraordinary waves is also considered. In a three-dimensional inhomogeneous ionosphere, stationary in time (N/t = 0), is determined in the general case, by an equation with three variables. The equation for proper depends only on the local value of the electron concentration N _c around the radiator and on integral values, determining, by means of additional calculations, the angle of refraction or its components, the horizontal gradients of electron concentration N/x and N/y, and in some cases, the integral electron concentration ₀ ^zcN dz. We describe the analysis of the measurements, made with the satellites Cosmos I, II and partially XI, assuming that N/t = N/y = 0, with a two variables equation. The expected errors are considered. The results coincide well for different points (Moscow, The Crimea, Sverdlovsk) and thus agree with the measurements of _H and with height-frequency ionospheric characteristics. The curve giving electron concentration versus height N (z) in the outer ionosphere (above the maximum of F2), shows a new maximum higher than the main maximum of the ionosphere N _MF2 at 120–140 km. At this maximum the value of N (z) is (0.9–0.95) N _MF2. The new data on the large-scale horizontal inhomogeneities of the ionosphere, exceed the previous ones by about a factor 10. By means of the irregular variations of the spectrum W() of the inhomogenous formation is determined. Three unknown constant maxima with values 16 to 18 km, 28 to 32 km and 100 to 120 km are found. The spectrum W () mainly characterizes the local properties of the ionosphere along the orbit of the satellite.     

Energetic particles at high latitudes

We review work on diffusion coefficients of energetic particles with an attempt to extract implications on their behaviour at high latitudes. In the ecliptic plane results from solar energetic particle propagation between the Sun and about 5 AU can be described by an effective radial mean free path _r which is approximately constant as a function of distancer. When particle propagation in three dimensions in the heliosphere is considered it is not sufficient to consider _r only. Jovian electrons can be used as probes to determine the parameters of three-dimensional diffusion. In the polar regions diffusion is dominated by its parallel component. Some predictions how should vary with latitude are discussed. For different choices of this variation we present expectations for intensity-time profiles of solar particle events during the Ulysses polar passages.     

The effect of rotation on the hydrodynamics of stellar collapse

Summary Using values of d, _min, and _max that Van Riper (1978) has found most promising for a hydrodynamic envelope ejection, we have shown that even a small amount of rotation in the initial core can stop its collapse before nuclear densities are reached. We expected _i > 0.02 to produce significant deviations from a spherically symmetric collapse, but have found that _i as much as ten times smaller than this will not allow the core to reach densities as high as in the spherical collapse. In no case, however, does the core flatten very much, nor does the value of become very large. Low final 's preclude the formation of an axisymmetric torus. They also indicate that deformation of an iron core into a triaxial configuration or fragmentation of the core during its collapse is an extremely unlikely event. (Note: Classically, must exceed 0.27 before a dynamic instability to non-axisymmetric perturbations is encountered.)The small degree of flattening of the core also suggests that the reduced moment of inertia I of the core will always be relatively small in magnitude and hence that the third time derivative of I, which is proportional to the energy emitted in gravity wave radiation, will not be very significant. Numerically calculated estimates of I- during some of these model evolutions supports this suspicion. If the _min and used here are found to be realistic values after the detailed physics of the core collapse is well understood, it is clear that gravitational radiation from a core collapse will be difficult to measure.Finally, we should point out that it is the relatively large values of Y_min (near 4/3) combined with values of d near unity that (a) prevented the core from flattening significantly in these models and (b) prevented the core from reaching high configurations. If realistic values of either one (or both) of these parameters are found to be much smaller in more complete models of the core collapse, then the core will have to become flatter (and denser) before pressure gradients will support it along the rotation axis. All of the conclusions drawn here would be modified accordingly under those circumstances. It should also be noted that in general relativistic models, the critical for spherical collapse is somewhat larger than 4/3 (Van Riper, 1979). Therefore, we predict that when fully general relativistic core collapses are performed including rotation, a given choice of _min and _i will produce a slightly flatter and slightly denser core than the corresponding model that has been presented here.     

Near Earth Current Meander (Necm) Model of Substorms

We propose that the appropriate instability to trigger a substorm is a tailward meander (in the equatorial plane) of the strong current filament that develops during the growth phase. From this single assumption follows the entire sequence of events for a substorm. The main particle acceleration mechanism in the plasma sheet is curvature drift with a dawn-dusk electric field, leading to the production of auroral arcs. Eventually the curvature becomes so high that the ions cannot negotiate the sharp turn at the field-reversal region, locally, at a certain time. The particle motion becomes chaotic, causing a local outward meander of the cross-tail current. An induction electric field is produced by Lenz's law, E ^ind=–A/t. An outward meander with B _z>0 will cause E×B flow everywhere out from the disturbance; this reaction is a macroscopic instability which we designate the electromotive instability. The response of the plasma is through charge separation and a scalar potential, E ^es=–. Both types of electric fields have components parallel to B in a realistic magnetic field. For MHD theory to hold the net E must be small; this usually seems to happen (because MHD often does hold), but not always. Part of the response is the formation of field-aligned currents producing the well-known substorm current diversion. This is a direct result of a strong E ^ind (the cause) needed to overcome the mirror force of the current carriers; this enables charge separation to produce an opposing electrostatic field E ^es (the effect). Satellite data confirm the reality of a strong E in the plasma sheet by counter-streaming of electrons and ions, and by the inverse ion time dispersion, up to several 100 keV. The electron precipitation is associated with the westward traveling surge (WTS) and the ion with omega () bands, respectively. However, with zero curl, E ^es cannot modify the emf =Edl=–d^M/dt of the inductive electric field E ^ind (a property of vector fields); the charge separation that produces a reduction of E must enhance the transverse component E . The new plasma flow becomes a switch for access to the free energy of the stressed magnetotail. On the tailward side the dusk-dawn electric field with EJ<0 will cause tailward motion of the plasma and a plasmoid may be created; it will move in the direction of least magnetic pressure, tailward. On the earthward side the enhanced dawn-dusk induction electric field with EJ>0 will cause injection into the inner plasma sheet, repeatedly observed at moderate energies of 1–50 keV. This same electric field near the emerging X-line will accelerate particles non-adiabatically to moderate energies. With high magnetic moments in a weak magnetic field, electrons (ions) can benefit from gradient and curvature drift to attain high energies (by the ratio of the magnetic field magnitude) in seconds (minutes).     

Co2 and Climate: a Geologist's View

Climate is discussed as an integral part of System Earth, determined by a complex interplay of numerous geological, biological and solar processes. The historical and geological record of changing climate and atmospheric CO₂ pressure does not support the current popular vision that this greenhouse gas is the dominant climate controlling agent. When empirically ante post tested against past global climate changes, the forecasts of the climate models mainly based on forcing by atmospheric CO₂ are not borne out. On the other hand, recent studies show that solar variability rather than changing CO₂ pressure is an important, probably the dominant climate forcing factor.     

Energy coupling between the solar wind and the magnetosphere

This paper describes in detail how we are led to the first approximation expression for the solar wind-magnetosphere energy coupling function , which correlates well with the total energy consumption rate U _T of the magnetosphere. It is shown that is the primary factor which controls the time development of magnetospheric substorms and storms. The finding of this particular expression indicates how the solar wind couples its energy to the magnetosphere; the solar wind and the magnetosphere constitute a dynamo. In fact, the power P generated by the dynamo can be identified as by using a dimensional analysis. Furthermore, the finding of indicates that the magnetosphere is closer to a directly driven system than to an unloading system which stores the generated energy before converting it to substorm and storm energies. Therefore, the finding of and its implications have considerably advanced and improved our understanding of magnetospheric processes. The finding of has also led us to a few specific future problems in understanding relationships between solar activity and magnetospheric disturbances, such as a study of distortion of the solar current disk and the accompanying changes of . It is also pointed out that one of the first tasks in the energy coupling study is an improvement of the total energy consumption rate U _T of the magnetosphere. Specific steps to be taken in this study are suggested.     

The synthesis of helium and CNO isotopes in massive stars

We present helium and CNO isotopic yields for massive mass-losing stars in the initial mass range 15M M _i 50M . We investigate their dependence on assumptions about mass loss rates, internal mixing processes, and metallicity, and specify the contributions from stellar winds and from supernova ejecta.     

Some problems of gamma-astronomy

The modern state of gamma-ray astronomy is reviewed, the paper being mainly devoted to the theoretical models that describe generation of gamma-ray emission under astrophysical conditions. Basic information on the processes of generation and absorption of gamma-rays, as well as the results of observations for various gamma-ray photon energies are reported.In the region of soft gamma-ray emission (i.e., for energies less than tens of MeV), where emission in gamma-ray lines dominates, we also discuss the nature of gamma-ray bursts, the origin of gamma-ray emission from the galactic centre, etc.Discrete sources and, in particular, the mysterious source Cyg X-3 are discussed in the region of very high (E > 10¹² eV) and ultra-high (E > 10¹⁵ eV) energy gamma-ray emission.A larger portion of the review is devoted to the analysis of cosmic-ray origin on the basis of the available gamma-ray data in the region from several tens of MeV to several GeV. The peculiarity of this energy range is, in particular, in the fact that the diffuse galactic emission was observed mainly there. We also discuss the problem of determination of the cosmic-ray density gradient from the gamma-ray data.The origin of high-latitude gamma-ray emission, the problem of galactic gamma-ray halo, etc., are discussed.The theoretical models explaining the nature of unidentified gamma-ray sources, as well as the results of measurements and theoretical estimations of a gamma-ray flux from SN1987A are analysed.List of Notations m electron mass, m = 9.108 × 10^–28 g, - M proton mass, M = 1.672 × 10^–24 g, - e electron charge, e = 4.803 × 10^–10 CGS - c velocity of light, c = 2.9979 × 10¹⁰ cm s^–1, - k Boltzmann constant, k = 1.380 × 10^–16 erg grad^–1, - e electron - p proton - gamma-ray photon - p antiproton - ^{0 0}-meson - -lepton - e ⁺ positron - r, , x radio-frequency, gamma-ray, and X-ray emission bands - E total energy of a particle - E _k kinetic energy - p particle momentum - spectral index for particles - spectral index for emission - n particle density (concentration) - H magnetic field strength - T temperature - _ph energy of low-energy photons - emission frequency - r _H Larmor radius of relativistic particles - k wave number - , z cylindric coordinates, in this case the coordinate (radius) along the galactic disk, z perpendicular to the galactic disk - M solar mass, M = 1.99 × 10³³ g.     

LISM structure — Fragmented superbubble shell?

Small scale structure in local interstellar matter (LISM) is considered. Overall morphology of the local cloud complex is inferred from Ca II absorption lines and observations of H I in white dwarf stars. Clouds with column densities ranging from 2–100 × 10¹⁷ cm^–2 are found within 20 pc of the Sun. Cold (50 K) dense (10⁵ cm^–3) small (5–10 au) clouds could be embedded and currently undetected in the upwind gas. The Sun appears to be embedded in a filament of gas with thickness 0.7 pc, and cross-wise column density 2 × 10¹⁷ cm^–2. The local magnetic field direction is parallel to the filament, suggesting that the physical process causing the filamentation is MHD related. Enhanced abundances of refractory elements and LISM kinematics indicate outflowing gas from the Scorpius-Centaurus Association. The local flow vector and Sco data are consistent with a 4,000,000 year old superbubble shell at –22 km s^–1, which is a shock front passing through preshock gas at –12 km s^–1, and yielding cooled postshock gas at –26 km s^–1in the upwind direction. A preshock magnetic field strength of 1.6 G, and postshock field strength of 5.2 G embedded in the superbubble shell, are consistent with the data.Abbreviations LISM Local ISM - SIC Surrounding Interstellar Cloud - LIC Local Interstellar Cloud     

Cosmological variability of fundamental physical constants

Gamow was one of the pioneers who studied the possible variability of fundamental physical constants. Some versions of modern Grand Unification theories do predict such variability. The paper is concerned with three of the constants: the fine-structure constant , the ratio of the proton massm _p to the electron massm _e, and the ratio of the neutron massm _n tom _e. It is shown on the basis of the quasar spectra analysis, that all the three constants revealed no statistically significant variation over the last 90% of the life time of the Universe. At the 2 significance level, the following upper bounds are obtained for the epoch corresponding to the cosmological redshiftsz2–3: /<1.5×10^–3, m _p/m _p<2×10^–3, and m/m<3×10^–4, where x is a possible deviation of a quantityx from its present value,m=m _p+m _n, and the nucleon masses are in units ofm _e. (According to new observational data which became known most recently, m _p/m _p<2×10^–4) In addition a possible anisotropy of the high-redshift fine splitting over the celestial sphere is checked. Within the relative statistical error 3 < 1% the values of turned out to be the same in various quadrants of the celestial sphere, which corresponds to their equality in causally disconnected areas. However, at the 2 level a tentative anisotropy of estimated / values is found in directions that approximately coincide with the direction of the relic microwave background anisotropy.The revealed constraints serve as criteria for selection of those theoretical models which predict variation of ,m _p orm _n with the cosmological time.     

Fast Dust in the Heliosphere

The dynamics of dust particles in the solar system is dominated by solar gravity, by solar radiation pressure, or by electromagnetic interaction of charged dust grains with the interplanetary magnetic field. For micron-sized or bigger dust particles solar gravity leads to speeds of about 30 to 40 km s^–1 at the Earths distance. Smaller particles that are generated close to the Sun and for which radiation pressure is dominant (the ratio of radiation pressure force over gravity F _rad/F _grav is generally termed ) are driven out of the solar system on hyperbolic orbits. Such a flow of -meteoroids has been observed by the Pioneer 8, 9 and Ulysses spaceprobes. Dust particles in interplanetary space are electrically charged to typically +5 V by the photo effect from solar UV radiation. The dust detector on Cassini for the first time measured the dust charge directly. The dynamics of dust particles smaller than about 0.1 m is dominated by the electromagnetic interaction with the ambient magnetic field. Effects of the solar wind magnetic field on interstellar grains passing through the solar system have been observed. Nanometer sized dust stream particles have been found which were accelerated by Jupiters magnetic field to speeds of about 300 km s^–1.     

Study of magnetospheric dynamics using energetic solar particles

Information can be obtained from energetic particle measurements through the chemical composition, energy spectrum, directional anisotropy, temporal and spatial intensity variations. This is equivalent to saying that there is a distribution functionf _k(p,r,t) wherek corresponds to thekth particle species of momentump at positionr and timet.Particle transport is described by the Boltzmann equation, and because the densities are generally low in the case of cosmic rays or energetic solar flare particles, collective transport effects can be neglected. In the absence of magnetospheric motion it is relatively easy to treat the problems of particle transport as simple propagation of charged particles in a stationary magnetic field configuration using, for instance, trajectory calculations in model fields. The method here is to use correlated measurements of the particle distribution at two points along a dynamic trajectory, and in this way to learn something about the geomagnetic field. This approach provides a good basis from which to study magnetospheric dynamics. If the magnetosphere moves, large scale electric fields, turbulent electromagnetic fields and sources and sinks affect the propagation of energetic particles considerably. These effects change the distribution functionf _k(p,r,t) and can thus be detected.In this paper, we shall show the importance of the single particle approximation (trajectories in a reference field) in forming the basis of our understanding of the quiet-time penetration of cosmic rays into the magnetosphere, we shall consider the steady dynamics such as wave-particle inter-action and field line reconnection, which is believed to exist nearly all the time, and finally we shall review the work which has been done in the much more complex and less well-understood field of impulsive dynamics such as geomagnetic storms and substorms. This last topic is only just beginning to be investigated in detail, and it is hoped that the study of impulsive dynamics, using energetic particles, may be as successful as the study of the quiet magnetosphere and the steady dynamics.     

Determination of the characteristics of the gamma-ray telescope Gamma-1

The Gamma-1 telescope has been developed through a collaboration of scientists in the USSR and France in order to conduct -ray astronomical observations within the energy range from 50 to 5000 MeV. The major characteristics of the telescope were established by Monte-Carlo simulations and calibrations made with the aid of electron and tagged -ray beams produced by an accelerator, and these have been found to be as follows: the effective area for photons coming along the instrument's axis varies from about 50 cm² at E = 50 MeV to approximately 230 cm² at E 300 MeV; the angular resolution (half opening of the cone embracing 68% events) is equal to 2.7° at E = 100 MeV, and 1.8° at E = 300 MeV; the energy resolution (FWHM) varies from 70% to 35% as the energy of the detected photons increases from 100 to 550 MeV; the telescope's field-of-view at the half-sensitivity level is 300–450 square degrees depending upon the spectrum of the detected radiation, and the event selection logic. Proceeding from the thus obtained characteristics it is demonstrated that a point source producing a photon flux J (E 100 MeV) = 3 × 10^-7 cm^-2 s^-1, can be detected with a 5 significance by observing it during 10⁶ s at the level of the Cygnus background, and a source having intensity J (E 100 MeV) = 10^-6 cm^-2 s^-1 can be detected to within a mean square positional accuracy of about 15.     

Probing the solar wind acceleration region using spectroscopic techniques

Measurements of the intensities and profiles of UV and EUV spectral lines can provide a powerful tool for probing the physical conditions in the solar corona out to 8 R and beyond. We discuss here how measurements of spectral line radiation in conjunction with measurements of the white light K-corona can provide information on electron, proton and ion temperatures and velocity distribution functions; densities; chemical abundances and mass flow velocities. Because of the fundamental importance of such information, we provide a comprehensive review of the formation of coronal resonance line radiation, with particular emphasis on the H i L line, and discuss observational considerations such as requirements for rejection of stray light and effects of emission from the geocorona and interplanetary dust. Finally, we summarize some results of coronal H i L and white light observations acquired on sounding rocket flights.Paper presented at the IX-th Lindau Workshop The Source Region of the Solar Wind.     

Presupernova models and supernovae

Present status of the theories for presupernova evolution and triggering mechanisms of supernova explosions are summarized and discussed from the standpoint of the theory of stellar structure and evolution. It is not intended to collect every detail of numerical results thus far obtained, but to extract physically clear-cut understanding from complexities of the numerical stellar models. For this purpose the evolution of stellar cores is discussed in a generalized fashion. The following types of the supernova explosions are discussed. The carbon deflagration supernova of intermediate mass star which results in the total disruption of the star. Massive star evolves into a supernova triggered by photo-dissociation of iron nuclei which results in a formation of a neutron star or a black hole depending on its mass. These two are typical types of the sueprnovae. Between them there remains a range of mass for which collapse of the stellar core is triggered by electron captures, which has been recently shown to leave a neutron star despite oxygen deflagration competing with the electron captures. Also discussed are combustion and detonation of helium or carbon which take place in accreting white dwarfs, and the collapse which is triggered by electron-pair creation in very massive stars.Appendix: Notations A mass number of atomic nucleus - B _v(a, b) incomplete beta function - c _p specific heat at constant pressure - c _p sound velocity - c(sub) center of the star - E _e mean energy of an electron captured by nucleus - E _n nuclear energy release from unit mass of the nuclear fuel specified by n - E _thr threshold energy (9.3) - E _thr,0 energy difference between the ground states of daughter nucleus and parent nucleus (9.1) - E energy of gamma ray emitted from daughter nucleus (9.1) - E _v mean energy of a neutrino emitted by electron capture (9.1) - f flatness parameter (2.17) - g local gravitational acceleration (2.16) - H atomic mass unit - H _p scale height of pressure (2.22) - H (sub) hydrogen-burning shell - k Boltzmann constant - l mixing length of convection - L _cr(M _r ) local Eddington's critical luminosity (4.3) - L _n integrated nuclear energy generation rate by nuclear fuel specified by n - L _v neutrino luminosity - L _{v, cr}(M _r ) local Eddington's critical neutrino luminosity (11.2) - M (current) mass of a star - m _M core mass contained interior to the carbon-burning shell - M _Ch Chandrasekhar's limiting mass (9.6) - M _H core mass contained interior to the hydrogen-burning shell - M _He core mass contained interior to the helium-burning shell - M _ms mass of a star at its zero-age min-sequence - M _O core mass contained interior to the oxygen-burning shell - M _r mass contained interior to a shell at r - M _Si core mass contained interior to the silicon-burning shell - M _WD mass of white dwarf (7.1) - M ₀ normalization factor to the non-dimensional mass (3.3) - M ₁ core mass (3.6) - N polytropic index between pressure and density (2.3) - n polytropic index between pressure and temperature (10.1) - N _A Avogadro number - N _ad adiabatic polytropic index - N _e number of electrons in unit mass of matter - NSE nuclear statistical equilibrium - P pressure - ph (sub) photosphere - Q _e mass fraction of the envelope exterior of the shell e (2.14) - R stellar radius - r radial distance of a shell - r ₀ normalization factor to the non-dimensional radius (3.2) - s specific entropy - S _i specific entropy of ions - T temperature - U homology invariant defined by (2.1) - u _gas specific internal energy of gas - u _rad energy of the radiation field per volume in which unit mass of gas is contained (6.4) - V homology invariant defined by (2.2) - _def velocity of deflagration front (6.10) - X concentration by weight of hydrogen - Y concentration by weight of helium - Y _e mole number of electrons in one gram of matter (9.7) - Y _v mole number of neutrinos in one gram of matter - Z concentration by weight of the elements other than hydrogen and helium - z shock strength (6.6) - 1 (sub) usually denotes the core edge (2.13) - ratio of the mixing length to the scale height of pressure (l/H _p ) - ratio of gas pressure to the total pressure - ratio of the specific heats - gD locus of singularity in U-V plane (2.5) - M(H _p ) mass contained within unit scale height of pressure (4.4) - _ec energy rate by electron captures (9.5) - _n nuclear energy generation rate by the nuclear fuel specified by n - _v neutrino loss rate - L _v ^(D) neutrino loss rate excluding the neutrinos from the electron captures (9.4) - non-dimensional density (3.1) - P/, not the non-dimensional temperature (2.7) - _W Weinberg's angle (5.8) - opacity - _v neutrino opacity (11.2) - describes the effect of electron degeneracy in equation of state (2.19) - _ec rate of electron capture - mean molecular weight - _e mean molecular weight of electrons - _e chemical potential of an electron excluding the rest mass (8.1) - _i mean molecular weight of ions - non-dimensional radius (3.1) - non-dimensional pressure (3.1) - matter density - _cr ^GR critical density above which the general relativistic instability sets in - _cr critical density for reimplosion of the core by beta processes (Section 5) - _ign density at the ignition - _nse density above which the deflagrated matter results in NSE composition - _e non-dimensional entropy of electron-per one electron in units of k(9.2) - _ff timescale of free fall (6.2) - _h (H _p ) timescale of heat transport over unit scale height of pressure (4.4) - _n nuclear timescale for a change in temperature (6.1) - non-dimensional mass (3.1) - _e chemical potential of an electron in units of kT (8.1)     

Nonlinear theory of the two-point correlation function

We consider the influence of the nonlinear stage of gravitational instability on the two-point correlation functions of gravitationally bound objects. Based on the theory of nonlinear gravitational contraction of a single density peak of dissipationless matter (Gurevich and Zybin, 1988a,b; 1990) we develop a method for calculating the two-point correlation functions of different objects of any mass. The method works good in the region of strong correlations and can be easily extended to calculate higher correlation functions. We show that the main contribution to the correlation function _i in the region of strong correlations _i 1 is made by pair systems located outside large clusters of objects. In this region the shape of _i is determined only by the nonlinear dynamics of gravitational contraction of dissipationless matter and has the form _i C ^–, where 1.8 is a universal parameter.